1、常 用 积 分 公 式 （一）含有 的积分( ) ax b+ 0a 1dxax b+1ln ax b Ca+ + 2 ()ax b x+d11()(1)ax b Ca+ +（ 1 ） 3 dxxax b+21(ln)ax b b ax b Ca+ + 42dxxax b+22311()2()ln2ax b b ax b b ax b Ca+ + +5d()xx ax b+1lnax bCbx+ + 62d()xx ax b+21lnaaxbCbx b x+ + 72d()xxax b+21(ln )bax b Caax+ +822d()xxax b+231(2ln )bax b b ax b
2、Caax+ + 92d()xx ax b+211ln()ax bCbax b b x+（二）含有 ax b+ 的积分 10 dax b x+32()3ax b Ca+ 11 dx ax b x+322(3 2 ) ( )15ax b ax b Ca + 122dx ax b x+22 2 332(15 12 8 ) ( )105a x abx b ax b Ca+ + 13 dxxax b+22(2)3ax b ax b Ca+ 1142dxxax b+22 232(3 4 8 )15a x abx b ax b Ca + 15dxx ax b+1ln ( 0)2arctan ( 0)ax b
3、 bCbbaxbbax bCbbb+ + )222dxax b+1arctan ( 0)1ln ( 0)2axC bbabax bCbab ax b+ )292dxax bx c+2222222arctan ( 4 )4412 4ln ( 4 )42 4ax bCbac b ac bax b b acCb acbacaxbbac+ac302dxxax bx c+221dln22bxax bx ca a ax bx c+ +（六）含有22x a+ (0a )的积分 3122dxx a+1arshxCa+ 22ln( )x xa C+ 32223d()xxa+22 2xCaxa+3322dxxxa
4、+22x aC+ + 34223d()xxxa+221Cxa +335222dxxxa+222 22ln( )22xax axxa+ + + +C 362223d()xxxa+2222ln( )xx xa Cxa +3722dxx xa+221lnxaaCax+ 3822 2dxx xa+222xaCax+ + 3922dx ax+222 22ln( )22xax axxa+ + + +C 40223()dx ax+2222 4 223(2 5 ) ln( )88xx axa a xxa C+ + + + + 4122dx xa+x2231()3x aC+ 4222 2dx xa+x422 2
5、2 22(2 ) ln( )88xax axa xxa C+ + + + + 4322dxaxx+2222lnxaax aa Cx+ + 44222dxaxx+2222ln( )xax xa Cx+ + （七）含有22x a (0a )的积分 4522dxx a1archxxCxa+ =22ln x xa C+ 4622d()x3x a22 2xCaxa +4722dxxxa22x aC + 448223d()xxxa221Cxa +49222dxxxa222 22ln22xax axxa+ + +C 502223d()xxxa2222lnxx xa Cxa +5122dxx xa1arcco
6、saCax+ 5222 2dxx xa222xaCax+ 5322dx ax222 22ln22xax axxa + +C 54223()dx ax2222 4 223(2 5 ) ln88xx axa axxaC + + + 5522dx xax2231()3x aC+ 5622 2dx xax422 22 22(2 ) ln88xax axa xxaC + + 5722dxaxx22arccosax aa Cx + 58222dxaxx2222lnxax xa Cx + （八）含有22ax (0a )的积分 5922dxax arcsinxCa+ 6022d()xax322 2xCaa x
7、+56122dxxax22axC + 62223d()xxax221Cax+63222dxxax222arcsin22xaxax Ca + + 642223d()xxax22arcsinxxCaax+6522dxx ax221lnaaxCax+ 6622 2dxx ax222axCax + 66722daxx222arcsin22xaax Ca+ +x68223()axdx2222 43(5 2 ) arcsin88xxaxax aaC + + 6922dx axx2231()3ax C + 7022 2dx axx422 22(2 ) arcsin88xaxx aax Ca + + 7122
8、daxxx2222lnaaxaxa Cx + 72222daxxx22arcsinax xCxa + （九）含有2ax bx c+(0a )的积分 732dxax bx c+21ln 2 2ax b a ax bx c Ca+ + + 742dax bx c x+224ax bax bx ca+ 2234ln 2 28ac bax b a ax bx c Ca+ + + + 752dxxax bx c+21ax bx ca+ 23ln 2 22bax b a ax bx c Ca + + + 762dxcbxax+212arcsin4ax bCabac +772dcbxaxx+223224ar
9、csin484ax b b ac ax bcbxax Caabac+ + +2782dxxcbxax+23212arcsin24baxbcbxax Caabac + + +（十）含有x ax b或 ()()x ab x的积分 79 dx axx b () ()ln( )xax bbaxaxbxbC + + +80 dx axbx () ()arcsinxa xax bbbx bxC + + 81d()(x)x ab x 2arcsinxaCbx+()ab 104dsinxab x+222222tan12lntan2xabbaCxbaabba+ + 22()ab 106dcosxab x+tan
10、12lntan2xababbaCabba xabba+ +22()ab113 arcsin dxxa22arcsinxx axCa+ + 114 arcsin dxx xa2222( )arcsin24 4xa xxaxCa+ 1152arcsin dxx xa322221arcsin ( 2 )39xxx aaxCa+ + 116 arccos dxxa22arccosxx axCa + 117 arccos dxx xa2222( )arccos24 4xa xxaxCa + 1182arccos dxx xa322221arccos ( 2 )39xxx aaxCa+ + 119 arc
11、tan dxxa22arctan ln( )2xax ax Ca + 120 arctan dxx xa221( )arctanxaax xCa+ 1212arctan dxx xa33222arctan ln( )366xxaax ax Ca+ + （十三）含有指数函数的积分 122 dxax1lnxaCa+ 123 edaxx1eaxCa+ 124 edaxx x21(1)eaxax Ca + 125 ednaxx x11eenax n axndx xxaa126 dxxax21ln (ln )xxxaaC + 127 dnxx ax11dln lnnx n xnx axaaa128 esin daxbx x 221e(sin cos )axabxbbxCab +129 ecos daxbx x 221e(sin cos )axbbxabxCab+ +10
